Method for incremental field-of-view-MR imaging

ABSTRACT

A method for producing an image of a volume of interest using a Magnetic Resonance Imaging (MRI) system comprising the steps of acquiring a plurality of under-sampled Magnetic Resonance (MR) data sets for a plurality of regions of the volume of interest along an axis of translation within the MRI system and reconstructing the image of the volume of interest using the respective under-sampled MR data sets.

BACKGROUND OF THE INVENTION

[0001] This invention relates generally to medical imaging. Moreparticularly, this invention relates to the acquisition of magneticresonance signals and reconstruction of images from samples of theacquired signals using a Magnetic Resonance Imaging (MRI) system.

[0002] Generally, imaging using a MRI system involves imaging a volumeof interest in a MRI scanner's usable volume. The usable volume isdefined as a contiguous area inside the patient bore of a MagneticResonance scanner and it can be rather limited in size. Typically, whenthe usable volume fails to cover an extended object, a method forexamining the whole volume containing the object employs repeatedexecutions of positioning and imaging a fraction of the whole volumewithin the scanner's usable volume. A subsequent assembling operationthen assembles or “stitches” the regional images together to produce afinal image of the whole volume of interest. Such an approach istypically challenged by the “stitching” artifact issue in whichresulting final images often suffer from distinctive artifacts at theboundaries of the “stitched” pieces. Existing techniques achieve correctcombination of regional images through full spatial encoding alongpatient table motion direction. They minimize “stitching” artifacts byusing slab selection profiles that are as rectangular as possible,and/or discarding image data near the boundaries. As a result, thesetechniques tend to be inflexible, require prolonged radio frequency (RF)excitation, and involve considerable acquisition efficiency degradation.

[0003] What is needed is an effective and efficient method for producingan image of a volume of interest using a MRI system, particularly withthe volume of interest that extends beyond the usable volume of the MRIsystem. What is further needed is a method for acquiring andreconstructing, in an “incremental field of view” fashion, data sets ofa volume of interest using a MRI system that relaxes the requirements ofslab selection profile and spatial encoding execution.

BRIEF SUMMARY OF THE INVENTION

[0004] A method for producing an image of a volume of interest using aMagnetic Resonance Imaging (MRI) system comprising the steps ofacquiring a plurality of under-sampled Magnetic Resonance (MR) data setsfor a plurality of regions of the volume of interest along an axis oftranslation within the MRI system and reconstructing the image of thevolume of interest using the respective under-sampled MR data sets.

BRIEF DESCRIPTION OF THE DRAWINGS

[0005] The features and advantages of the present invention will becomeapparent from the following detailed description of the invention whenread with the accompanying drawings in which:

[0006]FIG. 1 illustrates a simplified block diagram of a MagneticResonance Imaging system to which embodiments of the present inventionare useful;

[0007]FIG. 2 is a diagram showing aspects of a MRI system for use inconnection with embodiments of the invention;

[0008]FIG. 3 graphically illustrates a sequence of stepped translationsand image acquisitions useful in embodiments of the present invention;

[0009]FIG. 4 graphically illustrates a RF spatially selective excitationprofile useful in embodiments of the present invention;

[0010]FIG. 5 graphically illustrates a reconstruction weighting functionuseful in embodiments of the present invention;

[0011]FIG. 6 illustrates a matrix useful in deriving the reconstructionweighting function exemplified by FIG. 5; and,

[0012]FIG. 7 graphically illustrates a k-space sampling grid useful inembodiments of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

[0013]FIG. 1 illustrates a simplified block diagram of a system forproducing images in accordance with embodiments of the presentinvention. In an embodiment, the system is an MR imaging system whichincorporates the present invention. The MR system could be, for example,a GE-Signa MR scanner available from GE Medical Systems, Inc., which isadapted to perform the method of the present invention, although othersystems could be used as well.

[0014] The operation of the MR system is controlled from an operatorconsole 100 which includes a keyboard and control panel 102 and adisplay 104. The console 100 communicates through a link 116 with aseparate computer system 107 that enables an operator to control theproduction and display of images on the screen 104. The computer system107 includes a number of modules which communicate with each otherthrough a backplane. These include an image processor module 106, a CPUmodule 108, and a memory module 113, known in the art as a frame bufferfor storing image data arrays. The computer system 107 is linked to adisk storage 111 and a tape drive 112 for storage of image data andprograms, and it communicates with a separate system control 122 througha high speed serial link 115.

[0015] The system control 122 includes a set of modules connectedtogether by a backplane. These include a CPU module 119 and a pulsegenerator module 121 which connects to the operator console 100 througha serial link 125. It is through this link 125 that the system control122 receives commands from the operator which indicate the scan sequencethat is to be performed. The pulse generator module 121 operates thesystem components to carry out the desired scan sequence. It producesdata that indicate the timing, strength, and shape of the radiofrequency (RF) pulses which are to be produced, and the timing of andlength of the data acquisition window. The pulse generator module 121connects to a set of gradient amplifiers 127, to indicate the timing andshape of the gradient pulses to be produced during the scan. The pulsegenerator module 121 also receives subject data from a physiologicalacquisition controller 129 that receives signals from a number ofdifferent sensors connected to the subject 200, such as ECG signals fromelectrodes or respiratory signals from a bellows. And finally, the pulsegenerator module 121 connects to a scan room interface circuit 133 whichreceives signals from various sensors associated with the condition ofthe subject 200 and the magnet system. It is also through the scan roominterface circuit 133 that a positioning device 134 receives commands tomove the subject 200 to the desired position for the scan.

[0016] The gradient waveforms produced by the pulse generator module 121are applied to a gradient amplifier system 127 comprised of G_(x), G_(y)and G_(z) amplifiers. Each gradient amplifier excites a correspondinggradient coil in an assembly generally designated 139 to produce themagnetic field gradients used for position encoding acquired signals.The gradient coil assembly 139 forms part of a magnet assembly 141 whichincludes a polarizing magnet 140 and a whole-body RF coil 152. Volume142 is shown as the area within magnet assembly 141 for receivingsubject 200 and includes a patient bore. As used herein, the usablevolume of a MRI scanner is defined generally as the volume within volume142 that is a contiguous area inside the patient bore where homogeneityof main, gradient and RF fields are within known, acceptable ranges forimaging. A transceiver module 150 in the system control 122 producespulses that are amplified by an RF amplifier 151 and coupled to the RFcoil 152 by a transmit/receive switch 154. The resulting signalsradiated by the excited nuclei in the subject 200 may be sensed by thesame RF coil 152 and coupled through the transmit/receive switch 154 toa preamplifier 153. The amplified MR signals are demodulated, filtered,and digitized in the receiver section of the transceiver 150. Thetransmit/receive switch 154 is controlled by a signal from the pulsegenerator module 121 to electrically connect the RF amplifier 151 to thecoil 152 during the transmit mode and to connect the preamplifier 153during the receive mode. The transmit/receive switch 154 also enables aseparate RF coil (for example, a head coil or surface coil) to be usedin either the transmit or receive mode. It is to be appreciated that RFcoil 152 is configured to be operable for MRI scanning as describedbelow, in which a subject is translated on a positioning device alongthe z-axis. As used herein, “adapted to”, “configured” and the likerefer to mechanical or structural connections between elements to allowthe elements to cooperate to provide a described effect; these termsalso refer to operation capabilities of electrical elements such asanalog or digital computers or application specific devices (such as anapplication specific integrated circuit (ASIC)) that is programmed toperform a sequel to provide an output in response to given inputsignals.

[0017] The MR signals picked up by the RF coil 152 are digitized by thetransceiver module 150 and transferred to a memory module 160 in thesystem control 122. When the scan is completed and an entire array ofdata has been acquired in the memory module 160, an array processor 161operates to Fourier transform the data into an array of image data.These image data are conveyed through the serial link 115 to thecomputer system 107 where they are stored in the disk memory 111. Inresponse to commands received from the operator console 100, these imagedata may be archived on the tape drive 112, or they may be furtherprocessed by the image processor 106 and conveyed to the operatorconsole 100 and presented on the display 104. As will be discussed withreference to embodiments below, further processing is performed by theimage processor 106 that includes reconstructing acquired MR image dataaccording to embodiments described below.

[0018] In an embodiment of the present invention, a method for producingan image of a volume of interest using a Magnetic Resonance Imaging(MRI) system comprises the steps of acquiring a plurality ofunder-sampled MR data sets for a plurality of regions of the volume ofinterest along an axis of translation within the MRI system andreconstructing the image of the volume of interest using the respectiveunder-sampled MR data sets. As used herein, a volume of interest refersto a volume within subject 200 (FIG. 1) that is being examined. Thevolume is either a three-dimensional (3D) volume or alternatively atwo-dimensional (2D) slice. The embodiments described herein areapplicable to 3D volumes and 2D slices. Generally, a volume of interestfor purposes of the invention is part of the subject and may extendbeyond the usable volume of the MRI scanner, for example the back orleg, and therefore requires at least two or more scans if using aconventional MRI technique.

[0019] Referring to FIG. 2, application of a z-direction spatiallyselective excitation defines an imaged region 210 within the scanner'susable volume (within volume 142 of FIG. 1), and translation of thepositioning device 134 (FIGS. 1 and 2) along the z-axis and coordinatedMR data acquisition effect complete coverage of the examined subject 200with a plurality of acquired MR data sets. Each of the MR data sets isdefined by imaged region 210 at a given position of translation ofpositioning device 134. It is to be appreciated that a MRI scanner isdesigned to accomplish field homogeneity subject to other importantconsiderations such as openness, speed and cost.

[0020] In this embodiment of the present invention, each of the MR datasets is acquired with the presence of non-uniform spatial selectivityalong the translation axis. Non-uniform spatial selectivity is achievedby appropriate design of the RF excitation pulse and/or the RF coil. Ina first embodiment, the RF excitation pulse used for exciting imagedregion 210 of FIG. 2 is selected to have a non-uniform spatialselectivity, e.g., a RF excitation pulse that achieves a Gaussianprofile. Referring to FIG. 4, there is shown such a Gaussian profile.Alternatively in a second embodiment, the receive coil in the MRIsystem, such as RF coil 152 of FIG. 1, is designed to have non-uniformsensitivity so as to achieve a desired spatial selectivity. Examples ofRF coils having non-uniform sensitivity include surface coils andconventional birdcage coils that are adapted for the present imagingtechnique. The resulting non-uniform spatial selectivity achieved byeither a non-uniform excitation pulse or a non-uniform coil sensitivityresult in MR data that have additional spatial information embedded inthe regional images each obtained at a position along the translationaxis.

[0021] Referring further to FIG. 2, the translation direction is shownas being along the z-direction consistent with typical MRI systems thattranslate a subject along the z-axis. Referring to FIG. 3, steppedtranslation of positioning device 134 (FIG. 2) as a function of time isshown as u(t) and image acquisition intervals 310 are shown occurringafter each stepped translation of the positioning device.

[0022] As used herein, the term “under-sampled” refers to the conditionin which a given MR data set is the result of a k-space sampling withdensity along one or more k axis substantially lower than what isnormally required by an aliasing-free scan.

[0023] Reconstruction of the image of the volume of interest using theunder-sampled MR data sets is performed by weighting and summing aliasedregional images in a manner that substantially eliminates aliasing inthe image of the full volume of interest. Reconstruction methods usefulin embodiments of the present invention are derived as follows.

[0024] Suppose w(z) represents amplitude/phase effects of spatialselectivity due to RF excitation selectivity and/or receiver coilsensitivity. Ignoring relaxations, motion and coupling effects, dataacquired when an imaged object is displaced along z by u (throughtranslation of the positioning device 134 (FIG. 1)) as samples ofS_(u)(k_(x),k_(y),k_(z)), the Fourier transform of M(x,y,z−u)w(z):

S _(u)(k _(x) ,k _(y) ,k _(z))=∫∫∫M(x,y,z−u)w(z)e ^(−j2π(k) ^(_(x))^(x+k) ^(_(y)) ^(+k) ^(_(z)) ^(z)) dxdydz  (1)

[0025] where M(x,y,z) denotes what transverse magnetization an idealnon-selective excitation would induce if the object should be at u=0.

[0026] As Equation 1 and its transformed form (Parseval's theorem)indicate, the MR signal samples each explores information of M(x,y,z) ina localized space-frequency neighborhood where the energies of w(z+u)and FT{w*(z)exp(j2π(k_(x)x+k_(y)y+k_(z)z))} are concentrated (FT=Fouriertransform and *=complex conjugate). This leads to a concept ofspace-frequency domain sampling, a generalization to conventional MRI'sconcept of k space sampling (i.e., frequency domain sampling). ForEquation 1 in particular, a z−k_(z) plane sampling/coverage perspectiveis relevant: sensitivity profile w determines the shape of thespace-frequency neighborhood, and positioning device translation andz-gradient spatial encoding jointly define z−k_(z) traversing andsampling. FIG. 7 illustrates a rectangular z−k_(z) sampling grid that isrealized with even k_(z) sampling and constant positioning devicestepping: k_(z)=(m−ε)Δ_(kz) and u=(n−α)Δ_(z) (0≦ε,α<1 accommodateoffsets). The localizd space-frequency neighborhood is illustrated at700.

[0027] For this z−k_(z) sampling, Equation 1 is rewritten as:

S _(n)(k _(x) ,k _(y),(m−ε)Δ_(kz))=e ^(−j2π(m−ε)Δ) ^(_(kz)) ^((n−α)Δ)^(_(z)) ∫f _(k) _(x) _(,k) _(y) (z)g _(m,n) ^(*)(z)dz  (2)

[0028] where

f _(k) _(x) _(,k) _(y) (z)=e ^(j2επΔ) ^(_(kz)) ^(z) ∫∫M(s,y,z)e^(−j2π(k) ^(_(x)) _(x+k) ^(_(y)) _(y)) dxdy and

g _(m,n)(z)=e ^(−j2πmΔ) ^(_(kz)) ^(z) ∫∫w*(z+nΔ _(z)).

[0029] Developments based on the known frame theory show that if{g_(m,n)} a family of weighted Fourier harmonics, constitutes a frame,then M(x,y,z) may be reconstructed substantially without error as:$\begin{matrix}{{M\left( {x,y,z} \right)} = {\sum\limits_{n}{{M_{n}\left( {x,y,z} \right)}{h\left( {z + {\left( {n - \alpha} \right)\Delta_{z}}} \right)}}}} & (3)\end{matrix}$

[0030]  where h(z) is a reconstruction weighting function that ispre-derived from Δz, Δ_(kz), and w(z), and regional image M_(n)(x,y,z)is computed as: $\begin{matrix}{{M_{n}\left( {x,y,z} \right)} = {\sum\limits_{m}{\left( {\int{\int{{S_{n}\left( {k_{x},k_{y},{\left( {m - ɛ} \right)\Delta_{k\quad z}}} \right)}^{{j2\pi}{({{k_{x}x} + {k_{y}y}})}}{k_{x}}{k_{y}}}}} \right)^{{{j2\pi}{({m - ɛ})}}{\Delta_{k\quad z}{({z + {{({n - \alpha})}\Delta_{z}}})}}}}}} & (4)\end{matrix}$

[0031] Let λ denote the product of Δ_(kz) and Δ_(z). In an example casewhere w(z) is Gaussian, {g_(m,n)} constitutes a frame when λ<1.

[0032] In further embodiments, Equations 3 and 4 are extended toaccommodate w's that have x- or y-dependency, leading to consequent x-or y-dependency of the pre-derived h's. A more significant extensionlies in the fact that x and y may be further similarly treated (steppedtranslation in 3 dimensions).

[0033] The space-frequency perspective leads to one key insight on k_(z)sampling density: Δ_(kz)<1/Δ_(z) (or equivalently, λ<1) becomes therequirement for gradient-driven z-direction spatial encoding, which mayrepresent a significant relaxation compared to conventional MRItargeting an identical spatial resolution. This is because while thepresent imaging method's z-gradient must effect the same extent of k_(z)traversing, sampling density along k_(z) may be reduced (i.e., spacingbetween samples may be increased): from a value upper-bounded by1/(width of spatial selectivity profile) to a value upper-bounded by1/Δ_(z). In the example case of Gaussian sensitivity profile,Δ_(kz)<1/Δ_(z) is a sufficient and necessary condition for resolvingM(x,y,z) without aliasing.

[0034] Equations 3 and 4 define a reconstruction method for M(x,y,z)given z−k_(z) plane sampling and a selected h function. Reconstructionof the full FOV image in accordance with Equations 3 and 4 requirescomputing a simple summation of spatially weighted M_(n)(x,y,z)'s, theregional images. Reconstruction of the regional images is desirablycarried out on-the-fly, each, essentially a standard Fourier transformbased reconstruction, computed with fast Fourier Transform (FFT). Onesubtlety is that, rather than the conventional width of 1/Δ_(kz), eachregional image's domain of definition along z is (−∞, +∞). Therefore,each FFT result needs to be replicated along z, as far as itscorresponding weighting function extends, to form a correspondingregional image. The method of the present invention imposes norestriction on sampling along k_(x) or k_(y). In further embodiments,for example, Cartesian or alternatively spiral k_(x)−k_(y) sampling areused and the results are reconstructed accordingly.

[0035] Given Δ_(z) and the pre-derived reconstruction weighting functionh, Equations 3 and 4 allow analysis of the noise propagation fromacquired MR data points to reconstructed image pixels which is usefulfor signal to noise ratio (SNR) optimization. In particular, withknowledge of noise variance/covariance of the MR data, an analyticalexpression exists that explicitly predicts noise variance/covariance ofthe reconstructed image as a function of z. Assuming additive white datanoise with standard deviation σ_(data) for example, the noise standarddeviation of a pixel at z is expressed as:

σ_(pixel)(z)=σ_(data){square root}{square root over (Σ_(n) |h(z−nΔ_(z))|²)}/{square root}{square root over (total number of data points atone station)}  (5)

[0036] It follows that at the same spatial resolution and FOV_(z) ofNΔ_(z), when compared to a reference conventional scan (defined as oneusing a constant spatial selectivity of unit amplitude andgradient-driven z-encodes of 1/FOV_(z) sample spacing), $\begin{matrix}{\frac{S\quad N\quad R_{i\quad F\quad O\quad V}}{S\quad N\quad R_{r\quad e\quad f\quad e\quad r\quad e\quad n\quad c\quad e}} = \frac{1}{\sqrt{\sum_{n}\left| {h\left( {z - {n\quad \Delta_{z}}} \right)} \right|^{2}}\sqrt{\lambda \quad N}}} & (6)\end{matrix}$

[0037] where SNR_(iFOV) refers to the signal to noise ratio in theincremental imaging methods described with reference to embodiments ofthe present invention. The term {square root}{square root over (λN)}appears in the denominator reflecting the intrinsic SNR penaltyassociated with a λN-fold reduction in the number of averaged noisesamples. The other term in the denominator, {square root}{square rootover (Σ_(n)|h(z−nΔ_(z))|²)}, is a geometrical factor that is one maintarget when minimization of SNR penalty is pursued.

[0038] In the present invention, the reconstruction weighting functionh(z) is pre-computed based on Δ_(z), Δ_(kz), and w(z). The followingembodiments all target at deriving the reconstruction weighting functionh(z) to additionally provide a) a measure of residual aliasing, and b) acapability to flexibly trade off integrity from aliasing for robustnessagainst noise (i.e., to improve signal to noise ratio by controllablyaccepting some level of residual aliasing).

[0039] A basic method of deriving reconstruction weighting function h(z)is now described. With the z−k_(z) plane sampling grid illustratedabove, a regional image reconstructed from data collected at position l(l is the index for the position of the positioning device: position lcorresponds to u=(l−α)Δ_(z)) is expressed as (x- and y-dependencysuppressed for simplicity): $\begin{matrix}{{M_{l}(z)} = {\sum\limits_{p}{{w\left( {z + {\left( {l - \alpha} \right)\Delta_{z}} - {p/\Delta_{k\quad z}}} \right)}^{{- {j2\pi ɛ}}\quad p}{M\left( {z - {p/\Delta_{k\quad z}}} \right)}}}} & (7)\end{matrix}$

[0040] Assuming (−RΔ_(z),+RΔ_(z)) defines the region outside of whichthe magnitude of w(z) is negligible. It is noted that only at positionsn−R, n−R+1, . . . , n+R−1 and n+R will pick up MR signals that havecontributions from spins in region (−nΔ_(z)−Δ_(z)/2, −nΔ_(z)+Δ_(z)/2),and hence need be taken into account in the reconstruction of theregion. Let the reconstruction be a weighted superposition of aliasedimages obtained at positions n−R, n−R+1, . . ., n+R−1 and n+R:$\begin{matrix}{{\sum\limits_{l = {n - R}}^{n + R}{{h\left( {z + {\left( {l - \alpha} \right)\Delta_{z}}} \right)}{M_{l}(z)}}} = {\sum\limits_{l = {n - R}}^{n + R}{{h\left( {z + {\left( {l - \alpha} \right)\Delta_{z}}} \right)}{\sum\limits_{p = {- P}}^{P}{{w\left( {z + {\left( {l - \alpha} \right)\Delta_{z}} - {p/\Delta_{k\quad z}}} \right)}^{{- {j2\pi}}\quad ɛ\quad p}{M\left( {z - {p/\Delta_{k\quad z}}} \right)}}}}}} & (8)\end{matrix}$

[0041] where P is the minimum integer that is greater or equal to(2R+1/2)λ. For a weighted superposition of (aliased) regional images, orcomponent-coil images, to accurately reconstruct a full-FOV image,Σ_(l)h(z+(l−α)Δ_(z))M_(l)(z) should match M(z) as closely as possibleregardless of M(z)'s shape. The aliased terms in Equation 8 thereforemust be substantially negligible in magnitude. This, when expressed inmatrix form, translates to Ah=e₁, where

h=[h(z+(−R−α)Δ_(z))h(z+(−R+1−α)Δ_(z)) . . . h(z+(R−α)Δ_(z))]^(T),

e ₁=[1 0 . . . 0]^(T),

[0042] and A is a matrix fully defined by Δ_(z), Δ_(kz) and w(z) and isshown in FIG. 6. Solving equation Ah=e₁ with least squares for each zlocation in [0,Δ_(z)) generates a weighting function.

[0043] In effect, for each z location, ∥Ah−e₁∥, the norm of thedifference between Ah and e₁, serves as a measure of residual aliasinglevel, and should be of substantially insignificant amplitude to ensureaccurate reconstruction. Relaxing requirement on ∥Ah−e₁∥ however, ispractically desirable for robustness/SNR considerations. Rather thanusing a simple least squares as above, the following describes twoembodiments which derive h for less signal to noise (SNR) penalty whilemaintaining control over reconstruction accuracy.

[0044] A first embodiment for deriving h seeks a desired balance betweenresidual aliasing and image SNR. It is primarily applicable whenreconstruction operates in a regime where λ=Δ_(kz)Δ_(z)>1 andreconstruction without aliasing is not possible. In this case, h isdesirably derived to minimize ∥Ah−e₁∥+η=81 h∥, where η is a weightingreflecting relative interest on maximum signal-to-noise ratio (SNR)versus minimum residual aliasing. This derivation is formulated andsolved as a least squares problem.

[0045] A second embodiment for deriving h seeks to maximize SNR withinthe confinement of a chosen upper limit on residual aliasing level,reflecting a prioritization on imaging accuracy control. It is primarilyapplicable when reconstruction operates in a regime whereλ=Δ_(kz)Δ_(z)≦1 and reconstruction without aliasing is realizable. It isknown that as λ approaches 1 or as the z−k_(z) plane samplingneighborhood assumes an extremely elongated shape, reconstructionrobustness and SNR degrades. In the case of Gaussian profile, numericalstability of a reconstruction that strives for minimum aliasing worsensdrastically when λ goes beyond 0.996. For the regime, the presentembodiment formulates the derivation of h as a problem of finding the hthat minimizes ∥h∥ subject to μAh−e₁∥<τ, where τ is a scalarrepresenting a maximum level of tolerable residual aliasing. Singularvalue decomposition suggests itself as a powerful numerical tool. LetUΣV* denote the singular value decomposition of matrix A, where U and Vare orthogonal matrices and Σ is a diagonal matrix with σ_(j)'s, thesingular values of A, on the diagonal. A standard minimum norm solutionis found by computing VΣ^(†)U*e₁, where Σ^(†) is the result oftransposing Σ and replacing each of the non-zero σ_(j)'s with itsreciprocal. In the present embodiment, a threshold ρ is chosen, each ofthe 1/σ_(j)'s in Σ^(†) is set zero whenever σ_(j)≦ρ, and then VΣ^(†)U*e₁is computed. The result effectively minimizes image noise standarddeviation for a certain residual aliasing level. The flexibility intrading off reconstruction accuracy for lower image noise is appreciatedby noting the property that, if ρ₁≦ρ₂, then τ₁≦τ₂ but ∥h₁∥≧∥h₂∥. Therational behind the method include: a) the more zeros these 1/σ_(j)'sare replaced with, the smaller the norm of VΣ^(†)U*e₁, the solution forh, and typically the larger the norm of the error (residual aliasinglevel), and b) {square root}{square root over(Σ_(n)|h(z−nΔ_(z))|²)}=∥h∥.

[0046] Finally it can be shown that a useful scaling property holds. Ifh(z) denotes optimum reconstruction weighting function derived for agiven {w(z), Δ_(z) and Δ_(kz)} set using the methods described earlier,then for {w(β_(z)), βΔ_(z) and Δ_(kz)/β}, the optimum weighting ish(β_(z)).

[0047] Referring to FIG. 5, there is shown a representative illustrationof weighing h(z) are described above in Equations 3-8.

[0048] In a further embodiment, reconstruction computations usingEquation (3) are evaluated on-the-fly, with each M_(n)(x,y,z) computedwith fast Fourier Transform (FFT) techniques. As used herein,reconstructing “on-the-fly” refers to incrementally reconstructing animage with the evaluated Fourier transform in which the computationsoccur during data acquisition. Thus, reconstruction computation occurs“on-the-fly”, that is it is initiated and executed during the time ofdata acquisition by the MR scanner.

[0049] In further embodiments, reconstruction is based on adaptations ofknown techniques of filling up skipped k-space lines based onsynthesizing Fourier harmonics or known techniques of resolvinglocalization ambiguities directly with coil sensitivity mapping. In afirst further embodiment, reconstruction is based on filling up skippedk-space lines based on approximating Fourier harmonics with linearlycombined spatial selectivity profiles and thus reconstruct a full-FOVimage substantially free of aliasing artifacts. In a second furtherembodiment, reconstruction is processed in accordance with sensitivityencoding techniques. Such sensitivity encoding is based on resolvinglocalization ambiguities by algebraically extracting spatial informationencoded with the spatial selectivity profiles and thus reconstruct afull-FOV image substantially free of aliasing artifacts.

[0050] A further embodiment includes adapting the described embodimentsfor use in a conventional multi-slab volume imaging setting, in whichthe subject remains stationary and w(z) displaces incrementally toselect respective z-slabs which are then combined to form a full volume.Thus, in this alternative embodiment, the volume of interest is fixed inspace and the translation is translation of the excited locationrelative to the fixed volume of interest.

[0051] While the preferred embodiments of the present invention havebeen shown and described herein, it will be obvious that suchembodiments are provided by 9.;w way of example only. Numerousvariations, changes and substitutions will occur to those of skill inthe art without departing from the invention herein. Accordingly, it isintended that the invention be limited only by the spirit and scope ofthe appended claims.

What is claimed is:
 1. A method for producing an image of a volume ofinterest using a Magnetic Resonance Imaging (MRI) system comprising:acquiring a plurality of under-sampled Magnetic Resonance (MR) data setsfor a plurality of regions of said volume of interest along an axis oftranslation within the MRI system; and, reconstructing said image ofsaid volume of interest using said respective under-sampled MR datasets.
 2. The method of claim 1 wherein the acquiring step uses anexcitation pulse with non-uniform spatial selectivity along thetranslation axis.
 3. The method of claim 1 wherein the acquiring stepuses a receive coil with non-uniform sensitivity along the translationaxis.
 4. The method of claim 2 wherein said non-uniform spatiallyselective excitation pulse is a RF excitation pulse having a Gaussianprofile.
 5. The method of claim 3 wherein said receive coil withnon-uniform spatial sensitivity is a surface coil.
 6. The method ofclaim 3 wherein said receive coil with non-uniform spatial sensitivityis a birdcage coil.
 7. The method of claim 1 wherein said translation istranslation of said volume relative to an imaged location.
 8. The methodof claim 1 wherein said volume is fixed in space and said translation istranslation of spatial selectivity relative to said volume.
 9. Themethod of claim 1 wherein said axis of translation is the z-axis of theMRI system and said translation is stepped and synchronized with theacquiring step.
 10. The method of claim 1 wherein the reconstructingstep is performed by weighting and summing a plurality of regionalimages computed by said plurality of under-sampled MR data sets.
 11. Themethod of claim 1 wherein the reconstructing step comprises the stepsof: computing a plurality of regional images from said respectiveunder-sampled MR data sets; and, weighting and summing said plurality ofregional images to produce said image of said volume of interest. 12.The method of claim 1 wherein the reconstructing step is performedincrementally during the acquiring step.
 13. The method of claim 1wherein the reconstructing step is processed in accordance with spatialharmonic synthesizing techniques.
 14. The method of claim 1 wherein thereconstructing step is processed in accordance with sensitivity encodingtechniques.
 15. A method for producing an image of a volume of interestusing a Magnetic Resonance Imaging (MRI) system comprising: incrementinga positioning device along an axis of the MRI system through a pluralityof given positions to incrementally define a plurality of respectiveregions within said volume of interest; acquiring a respectiveunder-sampled MR data set for each of said respective plurality ofregions; computing a plurality of regional images corresponding to saidunder-sampled MR data sets for each of said given positions; and,reconstructing said image of said volume of interest from saidrespective regional images.
 16. The method of claim 15 wherein theacquiring step uses an excitation pulse with non-uniform spatialselectivity along the axis.
 17. The method of claim 15 wherein theacquiring step uses a receive coil with non-uniform sensitivity alongthe axis.
 18. The method of claim 16 wherein said non-uniform spatiallyselective excitation pulse is a RF excitation pulse having a Gaussianprofile.
 19. The method of claim 15 wherein the reconstructing stepfurther comprises the steps of: weighting and summing said plurality ofregional images to produce said image of said volume of interest. 20.The method of claim 15 wherein the reconstructing step is performedincrementally during the acquiring step.
 21. The method of claim 15wherein the reconstructing step is processed in accordance with${M\left( {x,y,z} \right)} = {{\sum\limits_{n}{{M_{n}\left( {x,y,z} \right)}{h\left( {z + {\left( {n - \alpha} \right)\Delta_{z}}} \right)}\quad {and}\quad {M_{n}\left( {x,y,z} \right)}}} = {\sum\limits_{m}{\left( {\int{\int{{S_{n}\left( {k_{x},k_{y},{\left( {m - ɛ} \right)\Delta_{k\quad z}}} \right)}^{{j2\pi}{({{k_{x}x} + {k_{y}y}})}}{k_{x}}{k_{y}}}}} \right)^{{{j2\pi}{({m - ɛ})}}{\Delta_{k\quad z}{({z + {{({n - \alpha})}\Delta_{z}}})}}}}}}$

 where h(z) is a reconstruction weighting function that is pre-derivedfrom Δ_(z), Δ_(kz), and w(z), w(z) being a spatial selectivity.
 22. Themethod of claim 21 wherein h(z) is derived in accordance with singularvalue decomposition techniques that minimize ∥h∥ subject to ∥Ah−e₁∥<τ,where τ is a scalar representing a maximum level of tolerable residualaliasing.